Caliban's Will puzzle

When Caliban's will was opened it was found to contain the following clause:

I leave ten of my books to each of A, B, and C, who are to choose in a certain order.

No person who has seen me in a green tie is to choose before A.

If B was not in Oxford in 1920, the first chooser never lent me an umbrella.

If B or C has second choice, C comes before the one who first fell in love.

Unfortunately A, B, and C could not remember any of the relevant facts; but the family solicitor pointed out that, assuming the problem to be properly constructed (i.e. assuming it to contain no statement superfluous to its solution) the relevant data and order could be inferred. What was the prescribed order of choosing; and who first fell in love?

If you tackle this, remember that every statement is necessary. Also remember that the requirement (that the puzzle represented by the three statements is properly constructed) is not part of the original puzzle -- it is a meta-statement about the base puzzle. [People regularly get confused about this point.] One further point you have to consider -- was the family solicitor correct in claiming (a) that the relevant data could be inferred and (b) that the order be inferred.

The puzzle was originally created by M.H. Newman, the exact wording was taken from Richard Harters' website, and I've discovered it from Anatoly Vorobey's blog.

Assuming the problem contains nothing superfluous:

- At least one person has seen Caliban in a green tie. So A is first or second chooser.
- B was not in Oxford in 1920, and the one of the candidates for first chooser lent Caliban an umbrella.
- B or C must be second chooser. So A must be first chooser.

This allows us to deduce that:

Then A is first chooser, C comes before the one who first fell in love (that must be B), so C is second chooser and B is third. A couldn't have been potentially switching with B since that would put A last; so A must have been potentially switching with C, but C lent Caliban an umbrella. B was the one who saw Caliban in a green tie.

So if A, B and C had remembered:

- B saw Caliban in a green tie, and is second or third.
- B was not in Oxford in 1920, and A never lent Caliban an umbrella, so A is first.
- B or C are second, but B first fell in love so C comes before him.

• I don't understand your first deductions. "If B was not in Oxford in 1920, the first chooser never lent me an umbrella." B may have been in Oxford in 1920, and, therefore, the first chooser DID lend him an umbrella. (Perhaps that is the relevance -- the fist chooser is the person who has lent him an umbrella.) The last statement I have the same problem with. Jan 17, 2015 at 20:36
• @EFrog I believe you are interpreting the statement as "If and only if B was not in Oxford...". I am reading it analogously to "If you are Kermit, you are green"; if you are not Kermit, you may or may not be green. Jan 18, 2015 at 4:27
• So you don't really need the "If B was in Oxford..." statement, right? "No person who has seen me in a green tie is to choose before A." implies that A is 1 or 2, then "If B or C has second choice, C comes before the one who first fell in love." implies that B or C is 2 (so A is 1) and then that same statement says that actually C must be 2 (so he can come before someone). Jan 19, 2015 at 16:45
• @Quinn you don't need that statement if you assume all the statements are relevant. But if you can't assume that and do remember who did what, then you need that statement. Caliban intended you to need it rather than just assume statement 3 applied. Jan 19, 2015 at 20:18
• I agree with @Quinn. If you assume the first and third statements must be relevant, you logically conclude the order must be ACB. The second statement therefore tells you nothing you don't already know regarding the order. Jan 21, 2015 at 16:11

I got this puzzle from "Can You Solve My Problems?" by Bellos and finding I didn't agree with his solution searched the web, all solutions examined seemed to match Bellos but didn't notice how they made a statement superfluous which I shall explore below.

I leave ten books to each of Low(A), Y.Y(B) & Critic(C), who are to choose in a certain order:

S1. No person who has seen me in a green tie is to choose before A.
S2. If B was not in Oxford in 1920 the first chooser never lent me an umbrella.
S3. If B or C has second choice C comes before the one who first fell in love.

PRECEPTS:

P1: Caliban knows the order & makes his statements accordingly.
P2: Any conditions of an if clause may be false.

CONCLUSIONS:

C1. If neither B or C saw him in green tie then S1 is superfluous. So A is 1 or 2.

AA- (1)
---
---


C2. If B was in Oxford S2 is superfluous, it doesn't identify anybody. If B was not in Oxford it identifies chooser 1 as B, or one of A or C, i.e. B is 2 or 3.

AA- (2)  AA- (3)
B--      -BB
---      ---

(2) is not possible because it makes S3 superfluous. So B is 2 or 3.


C3. If B is 2 then C must be 1, to be before one who first fell in love.

AA- (4) Not possible, nobody can be 3.
-B-
C--

If C is 2 then B is either 1 or 3

AA- (5) or AA- (7)
B--        --B
-C-        -C-

(5) is not possible, nobody can be 3.
(7) would make S2 superfluous, we would know all choosers.
NOTE. (7) is in fact the popular solution, which we have proved can
be arrived at without using S2.

Therefore P2 is confirmed and neither B nor C are 2, i.e. A is 2,
and with (3) we conclude:

-A- (8)
--B
C--


C is chooser 1, and never lent him an umbrella, and never saw him in green tie. A is chooser 2. B is chooser 3, and saw him in a green tie.

• I apologise, after further pondering I realised that my previous solution is flawed. I worked on the basis of a statement being true or false. So if [3] is true then there is only one result A, C, B making [3] superfluous. What I needed to do was work on basis that Caliban wrote will thinking A,B and C would know historic facts and work out will. It is necessary to consider options for [3] true and false. Dec 18, 2021 at 15:54